Solute transport through radial advection–dispersion is a common phenomenon in many applications, such as aquifer decontamination, heat exchange for geothermal exploration, and tracer testing. Numerous analytical solutions to the problem of tracer testing in a radially divergent flow field are available for fitting breakthrough curves. However, all of these analytical solutions assume non-uniform flow velocity based on Thiem's solution, which varies only spatially, not temporally. Although an injection well with a constant injection rate in a tracer test causes both spatial and temporal variability of the radial flow field, previous analytical studies have employed a steady-state groundwater flow solution, i.e. Thiem's solution, to derive analytical solutions for breakthrough curves or spatial concentration distribution curves. To the best of our knowledge, no analytical solution to the radial advection–dispersion equation under spatially and temporally variable flow conditions driven by injection at a constant rate is currently available.

Here, we propose a novel semi-analytical solution for describing solute transport in spatially and temporally variable flow induced by an injection well with a constant injection rate. We find that, above certain threshold values of dimensionless parameters related to aquifer flow properties, solute transport parameters, and the injection rate, the differences between previous analytical strategies our proposed semi-analytical solution are negligible. These threshold values tend to increase more sharply with an instantaneous injection source than a continuous injection source. In other words, it is more difficult for an instantaneous injection source to reach the condition where the difference between our approach and previous ones is negligible than for a continuous injection source type. Our findings clarify when existing analytical solutions can be reasonably used for parameter estimation through fitting of the breakthrough curve to field data and when the proposed semi-analytical solution is a better alternative.

通过径向平流-扩散的溶质传输是许多应用中的常见现象，例如含水层去污、地热勘探的热交换和示踪剂测试。径向发散流场中示踪剂测试问题的许多分析解决方案可用于拟合突破曲线。然而，所有这些解析解都假设基于 Thiem 解的非均匀流速，该解仅在空间上变化，而不是在时间上变化。尽管在示踪剂测试中以恒定注入速率注入井会导致径向流场的空间和时间变化，但以前的分析研究采用稳态地下水流动解，即蒂姆解，来推导突破曲线的解析解或空间浓度分布曲线。

在这里，我们提出了一种新的半解析解，用于描述由具有恒定注入速率的注入井引起的空间和时间可变流动中的溶质传输。我们发现，在与含水层流动特性、溶质传输参数和注入速率相关的无量纲参数的某些阈值之上，我们提出的半解析解决方案与以前的分析策略之间的差异可以忽略不计。瞬时注入源比连续注入源的这些阈值趋向于急剧增加。换句话说，瞬时注入源比连续注入源更难达到我们的方法与以前的方法之间的差异可以忽略不计的条件。